PomeLemma 11.37.1.2E4E82F8C3FC For every $\nu \in \mathcal P(X)$,
\[ \operatorname {Ent}_{\mathrm{d}V}(\nu ) =\sup _{F\in C(X)}\left(\int _ XF\, \mathrm{d}\nu -\log \int _ X\mathrm{e}^ F\, \mathrm{d}V\right). \]
Lemma 11.37.12E4E82F8C3FC
Proof. If $\nu $ is not absolutely continuous, choose a compact set $K$ with $\mathrm{d}V(K)=0$ and $\nu (K)>0$. For $t>0$, regularity gives a continuous cutoff $0\leq \chi _ t\leq 1$, equal to one on $K$, whose support has volume at most $\mathrm{e}^{-2t}$. For $F=t\chi _ t$, the expression in the supremum is at least
\[ t\nu (K)-\log \bigl (\mathrm{d}V(X)+\mathrm{e}^{-t}\bigr ), \]
which tends to infinity with $t$.
Suppose now that $\nu =h\, \mathrm{d}V$. If its entropy is finite, Jensen's inequality for the logarithm gives
\[ \int _ XF\, \mathrm{d}\nu -\operatorname {Ent}_{\mathrm{d}V}(\nu ) \leq \log \int _{\{ h>0\} }\mathrm{e}^ F\, \mathrm{d}V \leq \log \int _ X\mathrm{e}^ F\, \mathrm{d}V. \]
For infinite entropy this upper bound is automatic. For the opposite inequality, set
\[ F_ k=\log \bigl (\max \{ k^{-1},\min \{ h,k\} \} \bigr ),\qquad k\geq 1. \]
Then $\int _ X\mathrm{e}^{F_ k}\, \mathrm{d}V\to 1$ and $\int _ XF_ k\, \mathrm{d}\nu \to \operatorname {Ent}_{\mathrm{d}V}(\nu )$, with the value $+\infty $ allowed: the positive parts of the integrands $hF_ k$ increase to that of $h\log h$, and their negative parts are dominated by the integrable negative part of $h\log h$. For each fixed $k$, approximate $F_ k$ in $L^1(\mathrm{d}V+\nu )$ by continuous functions with the same bounds; both integrals, including the exponential one, converge to those of $F_ k$. Hence the supremum over continuous functions is at least the value of the expression at $F_ k$, and letting $k\to \infty $ completes the proof. $\square$
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